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1200                           Appendix  VL

(8) Place a crank and a mass in such a position on the shaft as
to cause the moment shewn by the closing line of the moment
polygon. As this must be done for revolving and reciprocating
masses separately, we may have to close four polygons, and there
might be altogether four cranks witb their allied revolving and
reciprocating masses respectively.

To reduce this complication, place the reference plane across
the centre line of one of the original cranks, and let the angle of
that crank with its mass be determined by the closing line of the
force polygon for the other cranks. Also so arrange the other
masses that their moment polygon will close. We thus have no
additional crank to provide for, say, the revolving weights.

Next, let there be a definite and equal proportional relation
between tbe masses of the reciprocating and revolving weights
throughout, when each pair, though of both kinds, may be added
at their respective crank-pins, and the same pair of polygons
(force and moment) will serve for both sets of weights. This
extreme simplification will be difficult in some cases, but the
actual requirements for true balance, as they have been stated,
will be a sufficient guide for any case, however stubborn; indeed,
four-crank engines can always be thus simplified. It must also
be clear that the motion of the reciprocating masses has here
been considered as pure harmonic, and that the whipping of short
connecting rods -in high-speed engines would constitute an
additional lack of balance.

In 'further explanation let us take a suppositious example.
Four cranks, I, II, III, IV are placed in various positions on one
shaft, II and IV having known masses = i in each case, while
| of the mass on every crank is supposed to reciprocate. Then,

Mass moment of II = i x 3 = 3.

 IV = i * 7 = 7.

Setting these out in a moment polygon we find the closing line to
measure 7*6.

.'. Mass moment of III ==7-6

"and         Mass at III  -     1^52
t   Thus the mass moments  of I, II, III will balance on the